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Riemann theta functions

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1: 21.2 Definitions
β–Ί
§21.2(i) Riemann Theta Functions
β–ΊFor numerical purposes we use the scaled Riemann theta function ΞΈ ^ ⁑ ( 𝐳 | 𝛀 ) , defined by (Deconinck et al. (2004)), …Many applications involve quotients of Riemann theta functions: the exponential factor then disappears. … β–Ί
§21.2(ii) Riemann Theta Functions with Characteristics
β–Ί
§21.2(iii) Relation to Classical Theta Functions
2: 21.8 Abelian Functions
§21.8 Abelian Functions
β–ΊFor every Abelian function, there is a positive integer n , such that the Abelian function can be expressed as a ratio of linear combinations of products with n factors of Riemann theta functions with characteristics that share a common period lattice. …
3: 21.10 Methods of Computation
β–Ί
§21.10(i) General Riemann Theta Functions
β–Ί
§21.10(ii) Riemann Theta Functions Associated with a Riemann Surface
β–Ί
  • Deconinck and van Hoeij (2001). Here a plane algebraic curve representation of the Riemann surface is used.

  • 4: 21.9 Integrable Equations
    §21.9 Integrable Equations
    β–ΊTypical examples of such equations are the Korteweg–de Vries equation … β–Ίβ–Ί
    β–ΊSee accompanying textβ–Ί
    Figure 21.9.2: Contour plot of a two-phase solution of Equation (21.9.3). … Magnify
    β–Ί
    5: 21.3 Symmetry and Quasi-Periodicity
    β–Ί
    §21.3(i) Riemann Theta Functions
    β–Ί β–Ί
    §21.3(ii) Riemann Theta Functions with Characteristics
    β–Ί …For Riemann theta functions with half-period characteristics, …
    6: 21.1 Special Notation
    β–ΊUppercase boldface letters are g × g real or complex matrices. β–ΊThe main functions treated in this chapter are the Riemann theta functions ΞΈ ⁑ ( 𝐳 | 𝛀 ) , and the Riemann theta functions with characteristics ΞΈ ⁒ [ 𝜢 𝜷 ] ⁑ ( 𝐳 | 𝛀 ) . β–ΊThe function Θ ⁑ ( Ο• | 𝐁 ) = ΞΈ ⁑ ( Ο• / ( 2 ⁒ Ο€ ⁒ i ) | 𝐁 / ( 2 ⁒ Ο€ ⁒ i ) ) is also commonly used; see, for example, Belokolos et al. (1994, §2.5), Dubrovin (1981), and Fay (1973, Chapter 1).
    7: Sidebar 21.SB2: A two-phase solution of the Kadomtsev–Petviashvili equation (21.9.3)
    β–ΊSuch a solution is given in terms of a Riemann theta function with two phases. …
    8: 21.4 Graphics
    §21.4 Graphics
    β–ΊFigure 21.4.1 provides surfaces of the scaled Riemann theta function ΞΈ ^ ⁑ ( 𝐳 | 𝛀 ) , with … β–ΊFor the scaled Riemann theta functions depicted in Figures 21.4.221.4.5β–Ί
    β–Ί
    See accompanying text
    β–Ί
    Figure 21.4.4: A real-valued scaled Riemann theta function: ΞΈ ^ ⁑ ( i ⁒ x , i ⁒ y | 𝛀 1 ) , 0 x 4 , 0 y 4 . … Magnify 3D Help
    β–Ί
    β–Ί
    See accompanying text
    β–Ί
    Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: ⁑ ΞΈ ^ ⁑ ( x + i ⁒ y , 0 , 0 | 𝛀 2 ) , 0 x 1 , 0 y 3 . … Magnify 3D Help
    9: 21.5 Modular Transformations
    β–Ί
    §21.5(i) Riemann Theta Functions
    β–ΊEquation (21.5.4) is the modular transformation property for Riemann theta functions. β–ΊThe modular transformations form a group under the composition of such transformations, the modular group, which is generated by simpler transformations, for which ΞΎ ⁑ ( πšͺ ) is determinate: … β–Ί
    §21.5(ii) Riemann Theta Functions with Characteristics
    β–ΊFor explicit results in the case g = 1 , see §20.7(viii).
    10: 21.6 Products
    §21.6 Products
    β–ΊThen β–Ί
    21.6.3 j = 1 h ΞΈ ⁑ ( k = 1 h T j ⁒ k ⁒ 𝐳 k | 𝛀 ) = 1 π’Ÿ g ⁒ 𝐀 𝒦 𝐁 𝒦 e 2 ⁒ Ο€ ⁒ i ⁒ tr ⁑ [ 1 2 ⁒ 𝐀 T ⁒ 𝛀 ⁒ 𝐀 + 𝐀 T ⁒ [ 𝐙 + 𝐁 ] ] ⁒ j = 1 h ΞΈ ⁑ ( 𝐳 j + 𝛀 ⁒ 𝐚 j + 𝐛 j | 𝛀 ) ,
    β–ΊOn using theta functions with characteristics, it becomes … β–Ί
    §21.6(ii) Addition Formulas